By Arjeh M. Cohen

The 1st interactive direction masking first and moment yr algebra. ranging from such primary issues as integers and divisions, modular mathematics and polynomials the content material extends to jewelry, fields and permutation teams. The hypertext is written in Java-enhanced HTML, and Java applets illustrate the idea whereas additionally contributing interactive calculators for computing with integers, polynomials and diversifications. the pc algebra method hole is built-in all through, permitting the calculation and manipulation of mathematical gadgets. additionally, collections for Mathematica notebooks and Maple worksheets evaluation the algorithms provided. a number of selection workouts supply clients with rapid suggestions, whereas amenities for tracking scholars and a bulletin board entire this electronic direction.

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**Example text**

In other words, the products are equal. But the latter product equals ∏z∈Z/nZ× (a· z) = aΦ(p) · ∏z∈Z/nZ× z. By Invertibility of Products the product ∏z∈Z/nZ× z is invertible, so, multiplying both sides of the above equation by its inverse, we find aΦ(n) = 1. This proves the theorem. 11. The set Z/15Z× contains 8 elements, one of them being 7(mod 15). eqsmod 78 , 494 , 44 , 12 , 1, 15 This in accordance with Euler’s Theorem. Let n be an integer. The order of an element a in Z/nZ× is the smallest positive integer m such that am = 1.

Show that there exists a positive integer n such that a/pn is an integer and gcd (p, a/pn ) = 1. 32. Determine all primes less than 100. 33. Determine all primes of the form n3 + 1, with n an integer. 34. Which of the following integers is prime: 187, 287, 387, 487, or 587? 35. Let n be an integer greater than 1, and let p be the smallest divisor of n greater than 1. Prove that p is prime. 36. Determine the prime factorization of the integers 111, 143, 724, and 1011. 37. Prove that the cube root of 17 is not rational.

Let n be an integer. Consider Z/nZ, the set of equivalence classes of Z modulo n. Addition and multiplication with these classes can be defined in the following way. 39. Algebra Interactive Addition of congruence classes is defined in terms of representatives. For instance, to add the two congruence classes modulo 5 above take any representatives in each of these classes, say 6 in the first and 3 in the second. Then their sum, 9, is a representative of the sum of the two classes. 6 (Addition and Multiplication).

### Algebra Interactive by Arjeh M. Cohen

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